Migration of seismic data (moving reflectors in a subsurface image to their true depths by data processing) requires making some estimate of the acoustic velocity which normally varies throughout the subsurface region of interest. In this context, “velocity” normally means the velocity and also any anisotropy parameters included in the velocity model. It is well known that migration results depend on the assumed velocity in the subsurface and therefore in order to obtain the best possible image from seismic data the velocity has first to be estimated in an optimal manner. The existing technology typically uses ray tomography. In this case seismic reflectors are typically imaged at a range of offsets. (The offset is the source-receiver spacing). Each offset should in principle provide a similar image except for different seismic amplitudes. It is often noted, however, that the images at different offsets differ in the depth of each imaged reflector. This implies that the velocity is incorrect. The data are usually displayed as a gather at each of many image points, so that the seismic images are sequenced firstly at varying offsets and secondly over image location in horizontal image variable. In a typical tomography program the velocity is then adjusted in such a way as to flatten the gathers (i.e. to minimize the discrepancies in reflector depths among the members of the gather). The gathers may be offset gathers, angle gathers, shot gathers or receiver gathers. If the data are being processed with ray based techniques, it is common to use offset gathers.
In typical ray based tomography programs, the velocity inversion is usually unstable. Therefore it is common to add additional terms into the program in such a way as to stabilize the calculations. One effect of such terms is to reduce the spatial resolution. In addition, such terms are usually not related to the physics of the problem, and therefore create uncertainty about the reliability of the final velocity model.
Because the final seismic image depends to a considerable degree on the velocity model, it is important to obtain as accurate a velocity model as possible and the best possible spatial resolution. As noted in the references below, it has been found in refraction seismic work in whole earth geophysics that modeling the seismic arrivals using so-called fat rays produces more highly resolved images. However up to the present time there has not been any fat ray technique especially designed to apply to reflection images and at the same time containing the appropriate physics of the reflection imaging problem.
Dahlen et al. (2000a) show how to use fat rays in whole earth tomography using transmitted arrivals through the earth. While their paper contains a detailed mathematical analysis of the transmission imaging problem, they do not discuss the reflection imaging problem.
Dahlen et al. (2000b) provide a detailed discussion of P-wave ray tomography in a spherical earth model. For most imaging problems in reflection imaging, the velocity model is much more complex, and therefore new techniques are needed.
Cerveny and Soares (1992) describe how to construct fat rays for arbitrary velocity. Their methods are primarily analytical, and do not address the computational issues faced in typical reflection imaging situations. In particular, they assume that the fat rays are calculated by propagating energy from source to receiver. In actual practice, as disclosed in this application, it is more convenient and computationally more efficient to calculate the fat rays from each image point.
Xie and Yang (2008) and de Hoop et al (2006) discuss wave equation based velocity analysis. This approach has considerable potential advantages; however it typically requires very intensive computation especially for 3D processing which is the major application of velocity analysis methods.
Xu et al (2006) present a fat ray tomography technique. However their fat rays do not cover the physically correct zone in space, and therefore can be viewed as a way of improving the stability of the calculation without actually affecting the resolving power of the method.
Bevc et al (2008) present a wave equation based fat ray tomography method. Their fat rays also do not cover the region actually required even for the simplest earth models.
There is therefore a need for a technique that is mathematically stable, has optimal resolution and at the same time is acceptable in terms of the computational effort required. The present invention satisfies these criteria.